Roof construction

ABSTRACT

A roof comprises a ring conforming in plan to a closed curve, a plurality of sets of cables with the cables of each set extending in plan parallel to an axis of skewed symmetry of the closed curve, and a membrane attached to the cables which by means of inflation lifts the cables so as to pre-tension the roof and form a domed surface.

United States Patent Geiger Sept. 17, 1974 ROOF CONSTRUCTION 3.233.3762/1966 Naillon 52/80 [76] Inventor: Dav! Gelge" 788 Rverslde 3.744.1917/1973 Bi r ci 52/2 New York, NY. 10032 FOREIGN TEN S O C O PA T R APPL]ATI NS [22] 1973 225.411 12/1968 U.S.S.R 52/2 [21] Appl. No.: 322,531197.921 1/1966 ussR 1.037.628 7/1966 Great Britain .1 52/2 Related US.Application Data [63] Continuation of Scr. Nov 80.048. Oct. 12. 1970.

abandoned [30] Foreign Application Priority Data May 3, 1970 Japan4537445 [52] US. Cl 52/2, 52/80, 52/63 [51] Int. Cl E04b U345 [58] Fieldof Search 52/2, 80, 63, 83, 167

[56] References Cited UNITED STATES PATENTS 1,302,182 4/1919 Lanchcster52/2 2,355,248 8/1944 Stevens 52/2 Primary Examiner-Henry C. SutherlandAssistant E.\'ami11erHenry Raduazo Attorney, Agent. or FirmCurtis,Morris & Safford [57] ABSTRACT A roof comprises a ring conforming inplan to a closed curve, a plurality of sets of cables with the cables ofeach set extending in plan parallel to an axis of skewed symmetry of theclosed curve, and a membrane attached to the cables which by means ofinflation lifts the cables so as to pre-tension the roof and form adomed surface,

10 Claims, 12 Drawing Figures PAIENTEBSEH 1 1914 SHEET 1 OF 5 FIG. 1

6 X A R 0 v A M INVENTOR David H.Gei BYG ger A 6C1, WW TTORNEYS PAKNTED7 I974 SHEET 3 BF 5 m GE INVENTOR David H.Geiger BY $17 5: l

944 @464, ORNEYS PAIENIED 3.835.599

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FIG. 6

' INVENTOR David H. Geiger ZT T SQQS Pmmznsm 1 m4 sum 5 or 5 FIG. 9

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-INVENTOR David H. Geiger BY QM M @GMA/ ATTORNEYS ROOF CONSTRUCTION Thisis a continuation, of application Ser. No. 80,048, filed Oct. 12, 1970and now abandoned.

The invention relates to a roof construction and, more particularly, toa roof construction including a ring which restrains the remainingelements of the roof structure and in turn is subjected to internalstresses that are substantially compressive. The roof membrane issupported by means of a plurality of sets of cables, with the cables ofeach extending in plan parallel to an axis of skewed symmetry of thering. The cables of the two sets are clamped together at theirintersections and serve, by inflation of the space beneath them tosustain a membrane fastened to the cables, for example by means of websunder tension.

The invention will now be further described in terms of a presentlypreferred embodiment thereof and with reference to the accompanyingdrawings, in which:

FIG. 1 is a diagram of a closed curve to which the ring of the roofstructure of the invention may conform, and illustrating certainproperties of that curve useful in explaining the invention;

FIG. 2 is a diagram showing the family of superellipses (x/a) (y/b)" l,with a b but of the same values for all ellipses shown, and with mpassing through values (including non-integral values) from less thanunity up to infinity, for which latter value the ellipse takes the formof a rectangle circumscribing all ellipses of the family;

FIG. 3 is a further diagram useful in explaining the invention;

FIG. 4 is a plan diagrammatic view of the roof construction according tothe invention shown in vertical section in FIG. 5;

FIG. 5 is a sectional view taken on the line 55 in FIG. 4 and showing inaddition one form of supporting structure for the roof of FIGS. 4 and 5;

FIG. 6 is a sectional view at an enlarged scale through the ring of theroof of FIGS. 4 and 5 showing in addition the anchorage of a cablethereto and the fastening of the roof membrane thereto;

FIGS. 7 and 8 are views of clamping means to fasten together at theircrossing points cables of the two sets of parallel cables in the roof ofFIGS. 4 and 5; and

FIGS. 9 through 12 are diagrammatic representations of means forfastening the roof membrane to the ring and to the cables.

FIG. 1 shows two lines M and N that are not perpendicular which are axesof skewed symmetry for the closed curve 2 since any line L intersectingone of these axes (for example, line L intersecting axis N at B) isparallel to the other axis M and intersects the curve 2 at points C andA such that AB is equal to BC. In this manner closed curves having linesof skewed symmetry may be constructed. The curve 2 of FIG. 1 is shown byway of example, as an ellipse conforming to the usual equation (x/a)(y/b) l.

An example of a family of closed curves having this property is thefamily of superellipses (x/a)'" (y/b)" 1 shown in FIG. 2. Whatever thevalue of m, the ellipse can be circumscribed with a rectangle whosesides are perpendicular to the major and minor axes of the ellipse, thisrectangle being tangent to the ellipse at the intersections of the majorand minor axes with the ellipse. It is a property of the superellipsethat the diagonals 8 and of the circumscribing rectangle are its axes ofskewed symmetry, whatever the value of m.

This may be demonstrated by a coordinate transformation to the Q-vy axiswhere: x (m-Q) cos a. y (1 Q) cos a and a=tan' b/a.

In these expressions, illustrated with reference to FIG. 2, a is theangle between the major axis of the ellipses of that figure and theadjacent diagonal of the circumscribing rectangle. n is the lengthmeasured along the direction of one of those diagonals as a slantcoordinate axis, and g is the length measured along the other diagonalof the circumscribing rectangle as another slant cord axis.

With cables placed so as to project in plan as straight lines parallelto the axes of skewed symmetry of the closed curve, such as the axes Mand N of the ellipses of FIG. 2, the invention provides a roof structurein which the stresses are borne by a ring conforming, or conformingsubstantially, to the closed curve, which ring is substantially free ofbending moments in the horizontal plane and which is to that extentfunicular.

One roof construction according to the invention is shown in FIGS. 4 and5. A ring 20 of elliptical shape rests upon a suitable foundation shownillustratively in FIG. 5 as a berm or earthwork ledge 22, of the sameoutline. Two sets of intersecting cables 24 and 26 (FIG. 4) eachparallel in projection onto the horizontal plane to one of the diagonals8 and 10 of the circumscribing rectangle 12, support a membrane 28 (FIG.5) which may be, for example, of fabric or plastic. The mem brane coversthe space enclosed within the ring 20 and is fastened thereto and to thecables 24 and 26 by suitable means illustrated in FIGS. 6 and 9 to 12,to be described presently. In the embodiment of FIGS. 4 and 5 the roofis sustained by a supra-atmospheric pressure in the space 30 beneath it.A very small excess of pressure above atmospheric suffices for thispurpose and can be maintained with pumping means, ingress to and egressfrom the space 30 being had through a lock not shown in FIG. 5.

In accordance with the invention, the cables 24 and 26 are disposed toimpose upon the ring 20 substantially zero bending moments in thehorizontal plane. This is achieved by giving to the cables such lengthsthat with the roof load which they must support (whether an internal airpressure as in FIGS. 4 and 5, or a weight due to gravity), the cableswill assume a configuration in vertical planes such that the horizontalcomponent of the tensions therein at their anchor points with the ringwill be those required to impose on the ring a substantially funicularloading.

If as in FIG. 4 the cables are in two sets 24 and 26 of equally spacedcables parallel to the diagonal axes 8 and 10, the horizontal componentsof force to be exerted by each cable at its anchor points with the ringcan be evaluated in terms of the stress along, i.e., tangent to the axisof the ring (FIG. 3), at the intersections of the diagonals 8 and 10with the ring. Let this stress be denoted P an aggregate compressivestress in the ring on the assumption that the elements 24 and 26 arecables under tension.

With the cables parallel to the axes 8 and 10 of skewed symmetry, letthe horizontal component of the cable reactions H,- acting on the ringsegment EF of FIG. 2 extending from the intersection with the ring at Eof one diagonal 8 past the intersection therewith of the major axes tothe intersection therewith at F of the other diagonal 10 be as shown inFIG. 3. The forces P acting at E and F are the internal forces.Structural symmetry is assumed about the major and minor axes 4 and 6 ofthe ellipse to which the ring 20 conforms. By considering theequilibrium of segments of this free body from E to H E to H etc, thehorizontal reactions H H etc, can be determined in terms of P.Similarly, if there is considered the ring segment DE between theintersection with the ring at D of the opposite end of diagonal 10 andthe point E already defined, and if there is determined the relationshipbetween the horizontal component H, and the force P, one finds that theopposite ends of each cable in the set 26 exert the same horizontalreaction on the ring. This, however, is also the requirement forhorizontal equilibrium of the cable net when forces acting on the cablesare either vertical or symmetric with respect to a vertical axis ifsymmetry and the end points lie in the same horizontal plane.

With the horizontal components of tensions in the cables at their anchorpoints thus determined as a function of the axial force P in the ring atits intersections with the diagonals 8 and 10, a suitable shape for theroof in terms of the rise of the membrane above the plane of the ringand the consequent conformation of the cables themselves can be arrivedat by considering vertical equilibrium at each of the cable intersectionpoints. The material of the roof membrane and acces sory elements suchas the cables themselves being known, the weight per unit projected areacan be estimated. This gives the vertical force of gravity to besupported by air pressure within the enclosed space 30, re sulting in anet upward vertical force on the roof membrane. With this upward forceand with an assumed value of P, one can compute the rise in the cablesnec essary for the stress therein to have this vertical component and tohave simultaneously the desired horizontal component at the anchorpoints of the cables. If the resultant three-dimensional shape for theroof is not the one desired, another value of the axial ring stress Pcan be assumed and another shape can be computed for the roof by thesame process. In this way there can be found two roof shapes straddlingthe one desired, e.g. as to the degree of convexity thereof, andextrapolation between the two will yield the shape desired.

That is, one obtains a complete pattern or three dimensionalspecification of the geometry of the cable net the two sets of cablesplus the tension at both ends of each cable and hence moreover thetensions in the successive segments of each cable. From this informationit is possible for example to cut the cables and to mark them atintervals which will be spaced in service by the spacings which are toexist between adjacent intersections along each cable, due account beinghad in the process of marking for the differences between the uniformtension to which the cable is subjected in the shop and the non-uniformtensions along each cable in service, and due account being additionallyhad for other differences between shop and service conditions such astemperature.

The cables are thus available of suitable length and with markings attheir points of intended crossover one with another and with endmarkings (46 in FIG. 6) to be made to coincide with the working pointsmarked on the ring (45 in FIG. 6). The'cables can then be attached tothe ring and clamped together. The roof membrane is then applied and thedesign load is imposed by inflation. with the consequence that withinthe margin of error accepted, the final shape for the roof and theloading of the ring will be that intended in the design.

The membrane 28 is secured to each of the cables by means of a web 42and by means of lacing as shown in FIG. 6. The membrane is also securedto the ring by means ofa hold-down strip 44, suitably bolted orotherwise fastened through the membrane to the ring. This is also shownin FIG. 10, and FIGS. 11 and 12 illustrate a preferred method forattaching the webs 42 to the membrane.

The cables of the two sets are fastened to each other at theirintersections by means of clamps which may take the form shown in FIGS.7 and 8.

The clamp of FlGS. 7 and 8 comprises a pair of crossed split steel pipesections 50 and 51, the adjacent halves of the two sections beingoptionally welded to gether. Cables 24' and 26', one from each of thetwo sets of cables, lie in the split pipe sections which are heldtogether with adequate available force of friction by a pair of U-bolts52 and a plate 53. No interface material is provided between the cablesand the pipe sections for improved holding power.

Details of one form of ring 20 are shown in the crosssectional view ofFIG. 6. The entire roof, lightened by the upward force of inflation,rests on the ring 20 and thereby on the berm 22. The coefficient offriction between the ring and the berm must be of intermediate value,large enough to prevent the roof from sliding off but low enough topermit localized slippage of small ex tent rather than to cause failureto the berm. A single galvanized sheet as indicated at 33 in FlG. 6 on asmooth troweled concrete surface 32 of a ring-shaped slab resting on theberm is satisfactory.

The cables, of which one is shown at 40, pass through bores in the ringfor anchorage to the ring. Provision is made for adjustment in length ofthe cable so that an index point 46 on the cable, determined in thepreloading and marking process hereinabove described, will coincide withan index point 45 along the length of the bore. The index points 45 onthe ring lie, with respect to plan, on the closed curve to which thering basically conforms. Thus preferably this closed curve lies alongthe projection onto the horizontal of the center of gravity of thecross-sectional area of the ring.

I claim.

1. A non-circular and non-elliptical roof structure defining an enclosedoccupiable building space comprising a continuous ring projecting inplan to a closed curve having a plurality of skewed axes of symmetry, aplurality of sets of structural members operatively connected to saidring with the structural members of each set extending in plansubstantially parallel to a separate one of said skewed axes ofsymmetry, said structural members in one of said sets crossing thestructural members of another of said sets at predetermined locationsthroughout said structure and being operatively interconnected to eachother at substantially all of said crossing locations, and a membranesecured to said structural members whereby the stresses in said ring aresubstantially funicular for every selected closed curve in said familyof curves and the stresses in said structural members are non-uniform.

2. A roof structure as defined in claim 1 wherein said closed curve isdeveloped by use of the equation (x/a)" (y/bY" l where the value of m isselected as any number greater than 2.

3. A roof structure as defined in claim 1 wherein said skewed axes ofsymmetry coincide with the diagonals of the rectangle circumscribingsaid selected closed curve.

4. A roof structure as defined in claim 3 wherein said structuralmembers are tension cables.

5. A roof structure as defined in claim 4 including a foundationconforming in plan substantially to the closed curve defined by saidring and a support plate between said ring and foundation providing abearing surface frictionally engaged therebetween for supporting saidring on said foundation.

6. The roof structure as defined in claim 5 wherein said foundationincludes an earthern berm extending around the outside of said ring andbieng inclined outwardly and downwardly therefrom along an inclineconforming generally to the adjacent slope of the membrane to absorbpositive wind pressures.

7. An air-supported cable roof structure defining an enclosed occupiablebuilding space comprising, a peripheral support ring projecting in planto a closed curve having a plurality of skewed axes of symmetry anddeveloped by use of the equation (x/a)'" (y/b)"' 1, wherein m equals anynumber greater than 2 and the values of a and b are different; aplurality of sets of flexible tension cables operatively connected tosaid ring with the cables of each set extending in plan substantiallyparallel to a separate one of said skewed axes of symmetry, said cablesin one of said sets crossing the cables of another of said sets atpredetermined locations throughout said structure and being operativelyinterconnected to each other at substantially all of said crossinglocations, and a flexible membrane secured to said cables on the sidethereof opposite said occupiable building space thereby to enclose saidcables within said occupiable building space, said cables and membranebeing supported by pressurized air supplied to said structure, saidcables having non-uniform stresses therein under uniform load conditionsand said ring remaining in its closed curved configuration withoutdeformation while the stress in said ring are substantially funicular.

8. A roof structure as defined in claim 7 wherein said skewed axes ofsymmetry coincide with the diagonals of the rectangle circumscribingsaid selected closed curve.

9. A roof structure as defined in claim 8 including a foundationconforming in plan substantially to the closed curve defined by saidring, said ring including a support plate secured thereto and having alow friction surface engaged with said foundation for supporting saidring thereon and to permit controlled movement therebetween.

10. The roof structure as defined in claim 9 wherein said foundationincludes an earthen berm extending around the outside of said ring andbeing inclined outwardly and downwardly therefrom along an inclineconforming generally to the adjacent slope of the membrane to absorbpositive wind pressures

1. A non-circular and non-elliptical roof structure defining an enclosedoccupiable building space comprising a continuous ring projecting inplan to a closed curve having a plurality of skewed axes of symmetry, aplurality of sets of structural members operatively connected to saidring with the structural members of each set extending in plansubstantially parallel to a separate one of said skewed axes ofsymmetry, said structural members in one of said sets crossing thestructural members of another of said sets at predetermined locationsthroughout said structure and being operatively interconnected to eachother at substantially all of said crossing locations, and a membranesecured to said structural members whereby the stresses in said ring aresubstantially funicular for every selected closed curve in said familyof curves and the stresses in said structural members are non-uniform.2. A roof structure as defined in claim 1 wherein said closed curve isdeveloped by use of the equation (x/a)m + (y/b)m 1 where the value of mis selected as any number greater than
 2. 3. A roof structure as definedin claim 1 wherein said skewed axes of symmetry coincide with thediagonals of the rectangle circumscribing said selected closed curve. 4.A roof structure as defined in claim 3 wherein said structural membersare tension cables.
 5. A roof structure as defined in claim 4 includinga foundation conforming in plan substantially to the closed curvedefined by said ring and a support plate between said ring andfoundation providing a bearing surface frictionally engaged therebetweenfor supporting said ring on said foundation.
 6. The roof structure asdefined in claim 5 wherein said foundation includes an earthern bermextending around the outside of said ring and bieng inclined outwardlyand downwardly therefrom along an incline conforming generally to theadjacent slope of the membrane tO absorb positive wind pressures.
 7. Anair-supported cable roof structure defining an enclosed occupiablebuilding space comprising, a peripheral support ring projecting in planto a closed curve having a plurality of skewed axes of symmetry anddeveloped by use of the equation (x/a)m + (y/b)m 1, wherein m equals anynumber greater than 2 and the values of a and b are different; aplurality of sets of flexible tension cables operatively connected tosaid ring with the cables of each set extending in plan substantiallyparallel to a separate one of said skewed axes of symmetry, said cablesin one of said sets crossing the cables of another of said sets atpredetermined locations throughout said structure and being operativelyinterconnected to each other at substantially all of said crossinglocations, and a flexible membrane secured to said cables on the sidethereof opposite said occupiable building space thereby to enclose saidcables within said occupiable building space, said cables and membranebeing supported by pressurized air supplied to said structure, saidcables having non-uniform stresses therein under uniform load conditionsand said ring remaining in its closed curved configuration withoutdeformation while the stress in said ring are substantially funicular.8. A roof structure as defined in claim 7 wherein said skewed axes ofsymmetry coincide with the diagonals of the rectangle circumscribingsaid selected closed curve.
 9. A roof structure as defined in claim 8including a foundation conforming in plan substantially to the closedcurve defined by said ring, said ring including a support plate securedthereto and having a low friction surface engaged with said foundationfor supporting said ring thereon and to permit controlled movementtherebetween.
 10. The roof structure as defined in claim 9 wherein saidfoundation includes an earthen berm extending around the outside of saidring and being inclined outwardly and downwardly therefrom along anincline conforming generally to the adjacent slope of the membrane toabsorb positive wind pressures.